Very Hard Problem!!!!!!!

[Karnage]

Hey guys i had this problem that i want an explanation for. I want a good way to solve it probably using limits or some part of calculus. Can anyone find a good way ot figure it out?

 

Here's the problem: (TESTS YOUR INTELLIGENCE)

 

Two cities A and B are connected by a 100 mile railroad. In each city there is a train. At one instant the trains start moving toward each other with equal speed of 50 mph. Train A had a fly sitting on it, which took off at the moment of the train's departure and started flying toward train B with a speed of 90 mph. When the fly reached train B it turned around and started flying toward train A at 90 mph. The fly bounced between the trains multiple times, each time reversing its velocity, until the trains collided. What is the total distance flown by the fly?

 

Again, it's alright if you cant use calculus to solve this problem but I need a way to teach my classmates how to do it!!! I hope someone realy smart here can help me out! Thanks a lot for your time.

[BobbyJoeCool]

without putting too much thought into it... look at it this way...

 

the fly is moving at a constant speed. And if you mean distance traveled as in if I go a mile north and then a mile south did I travel 2 miles, or did I travel 0 (because I'm where I started...)

 

If the former, it a matter of finding at what time the trains collide, and then you have the speed of the fly, and you just calculate the distance traveled at a constant speed at time t (when the trains collide).

 

If it's the latter, I'd really have to think about it more.

[Karnage]

Can you elaborate please?

[BobbyJoeCool]

distance=speed*time.

 

figure out how long it will take for the trains to collide (they have to go a total of 100 miles, and have a combined speed of 100 mph... so they'll meet in 1 hour...)

 

t=1 hour

 

the speed of the fly is constant (because speed is the absolute value of velocity). First it moves at 90 MPH towards train one, and when it gets there, it's moving at the same speed back at train two, and back and forth with constant speed of 90 MPH...

 

s=90 miles/hour

 

d=s*t

 

d=90 miles/hour * 1 hour

 

d= 90 miles

[BobbyJoeCool]

As for the second way, if you're looking for the distance from the starting point that's easy... it's the point where the trains meet, which since they're going the same speed will be half way and hence 50 miles...

[Karnage]

OK thnx. I was hoping for a more complicated answer though

[shmoe]

There's an old joke about this problem (I modified the answer to match the version at hand):

 

----

When this problem was posed to John von Neumann, he immediately replied, "90 miles."

 

"It is very strange," said the poser, "but nearly everyone tries to sum the infinite series."

 

"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

----

 

The infinite series mentioned is the 'obvious' way of setting up the problem and computationally more work. Each term in the series is one leg of the flys journey. If you hope to teach a method like this to your classmates I'd suggest you try to set this sum up yourself, it's the best way to learn it before you attempt to teach it.

[Karnage]

What is the infinite series for this problem?